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arXiv:1605.05560 · IEEE International Black Sea Conference on Communications and Networking

Time-invariant spatially coupled low-density parity-check codes with small constraint length

Marco Baldi, Massimo Battaglioni, Franco Chiaraluce, Giovanni Cancellieri

Published Jan 1, 2016• 4 citations
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Abstract

We consider a special family of spatially coupled low-density parity-check (SC-LDPC) codes, that is, time-invariant low-density parity-check convolutional (LDPCC) codes, which are known in the literature for a long time. Codes of this kind are usually designed by starting from quasi-cyclic (QC) block codes, and applying suitable unwrapping procedures. We show that, by directly designing the LDPCC code syndrome former matrix without the constraints of the underlying QC block code, it is possible to achieve smaller constraint lengths with respect to the best solutions available in the literature. We also find theoretical lower bounds on the syndrome former constraint length for codes with a specified minimum length of the local cycles in their Tanner graphs. For this purpose, we exploit a new approach based on a numerical representation of the syndrome former matrix, which generalizes over a technique we already used to study a special subclass of the codes here considered.

DOI
10.1109/blackseacom.2016.7901543

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We consider a special family of spatially coupled low-density parity-check (SC-LDPC) codes, that is, time-invariant low-density parity-check convolutional (LDPCC) codes, which are known in the literature for a long time. Codes of this kind are usually designed by starting from quasi-cyclic (QC) block codes, and applying suitable unwrapping procedures. We show that, by directly designing the LDPCC code syndrome former matrix without the constraints of the underlying QC block code, it is possible to achieve smaller constraint lengths with respect to the best solutions available in the literature. We also find theoretical lower bounds on the syndrome former constraint length for codes with a specified minimum length of the local cycles in their Tanner graphs. For this purpose, we exploit a new approach based on a numerical representation of the syndrome former matrix, which generalizes over a technique we already used to study a special subclass of the codes here considered.
In this paper, we deal with time-varing quasi-cyclic (QC) spatially-coupled (SC) low-density parity-check (LDPC) codes by lifting the coupled protographs. The construction method of girth-6 protographs for any code rate is proposed. We provide a theoretical lower bound on the syndrome former constraint length under condition of the protographs without 4-cycles in their Tanner graphs. Then we use periodic QC lifting method to search the QC-SC-LDPC matrices with larger girth based on the protographs. The minimum dimension size of QC-SC-LDPC codes is given to count the girth of QC-SC-LDPC codes with a semi- infinite binary-check matrix structure. The performance of such constructed codes with lower complexity and delay is superior to the LDPC block codes.
In this paper, we propose a design technique for the construction of variable-regular time-invariant spatially-coupled low-density parity-check (SC-LDPC) codes with small constraint length and low error floor. The proposed technique reduces the error floor by imposing simple constraints on the short cycles in the code's Tanner graph, which in turn, result in the elimination of the most dominant trapping sets of the code. In some cases, we also derive lower bounds on the syndrome former memory for satisfying such constraints. The designed codes are superior to the state-of-the-art in terms of error floor performance and/or decoding complexity and latency.
Time-invariant spatially coupled low-density parity-check (TI-SC-LDPC) codes can be obtained by unwrapping quasi-cyclic (QC) LDPC block codes. This results in a free distance that is lower bounded by the minimum distance of the underlying QC-LDPC block codes. By introducing some variability in the syndrome former matrix, time-varying (TV) SC-LDPC codes are obtained, which trade an improved error correction performance for an increased decoding memory requirement and decoding complexity. A family of codes able to combine the advantages of TI-SC-LDPC codes with those of TV-SC-LDPC codes is that of periodically time-varying (PTV) SC-LDPC codes, based on a finite and periodic variation of the syndrome former matrix. In this paper we focus on such codes, and derive new upper bounds on the free distance of PTV-SC-LDPC code ensembles as well as on specific codes. By using these bounds, we show that PTV-SC-LDPC codes can achieve important improvements in the free distance over TI-SC-LDPC codes even using a very small period of variability, which corresponds to a minimal increase in memory and complexity. We also validate the new upper bounds through numerical experiments and assess the error correction performance of the corresponding codes through Monte Carlo simulations.
In this paper, time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs) are considered, and the connections existing between their low-weight codewords and cycles in their Tanner graphs are studied. Using the polynomial representation of these codes, we show that parity-check matrices having columns with weight ≥2 can be analyzed considering a certain number of parity-check sub-matrices having regular columns with weight 2. These sub-matrices are associated to cycles in the code Tanner graph and define as many codes we denote as component codes. Based on this observation, we find that codewords of the main code can be expressed as combinations of codewords of the component codes. The design of codes free of codewords up to a certain weight is also addressed. We show that low-weight codewords in the main code can be avoided by removing some types of cycles in its Tanner graph. Our design approach is applied to some well-known ensembles of SC-LDPC-CCs to prove its effectiveness.
We consider a special family of spatially coupled low-density parity-check (LDPC) codes known as LDPC convolutional codes and propose a design technique based on symbolic hyper-graphs aimed at achieving good codes with small syndrome former memory order. We define conditions on the minimum syndrome former memory order that must be imposed in order to avoid short local cycles in the codes Tanner graphs and to achieve good minimum distance. Code design examples are provided, accompanied by error rate performance assessments based on Montecarlo simulations.